Simplify the following expression: $x = \dfrac{9n^2 + 81n}{-81n^2 + 18n}$ You can assume $n \neq 0$.
Answer: Find the greatest common factor of the numerator and denominator. The numerator can be factored: $9n^2 + 81n = (3\cdot3 \cdot n \cdot n) + (3\cdot3\cdot3\cdot3 \cdot n)$ The denominator can be factored: $-81n^2 + 18n = - (3\cdot3\cdot3\cdot3 \cdot n \cdot n) + (2\cdot3\cdot3 \cdot n)$ The greatest common factor of all the terms is $9n$ Factoring out $9n$ gives us: $x = \dfrac{(9n)(n + 9)}{(9n)(-9n + 2)}$ Dividing both the numerator and denominator by $9n$ gives: $x = \dfrac{n + 9}{-9n + 2}$